Problem 26
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(y\) -axis. $$ y=9-x^{2}, \quad y=0, \quad x=2, \quad x=3 $$
Problem 26
Hydraulic Press In Exercises 25 and \(26,\) use the integration capabilities of a graphing utility to approximate the work done by a press in a manufacturing process. A model for the variable force \(F\) (in pounds) and the distance \(x\) (in feet) the press moves is given. $$ F(x)=\frac{e^{x^{3}}-1}{100} \quad 0 \leq x \leq 4 $$
Problem 27
In Exercises \(27-34,\) (a) use a graphing utility to graph the region bounded by the graphs of the equations, \((b)\) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results. $$ f(x)=x\left(x^{2}-3 x+3\right), \quad g(x)=x^{2} $$
Problem 27
In Exercises 27 and \(28,\) find the center of mass of the point masses lying on the \(x\) -axis. $$ \begin{array}{l} m_{1}=6, m_{2}=3, m_{3}=5 \\ x_{1}=-5, x_{2}=1, x_{3}=3 \end{array} $$
Problem 27
(a) Use a graphing utility to graph the function \(f(x)=x^{2 / 3}\). (b) Can you integrate with respect to \(x\) to find the arc length of the graph of \(f\) on the interval [-1,8]\(?\) Explain. (c) Find the arc length of the graph of \(f\) on the interval [-1,8] .
Problem 27
In Exercises \(27-30\), use the disk or the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about each given line. \(y=x^{3}, \quad y=0, \quad x=2\) (a) the \(x\) -axis (b) the \(y\) -axis (c) the line \(x=4\)
Problem 27
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. Verify your results using the integration capabilities of a graphing utility. $$ y=\cos x, \quad y=0, \quad x=0, \quad x=\frac{\pi}{2} $$
Problem 28
In Exercises 27 and \(28,\) find the center of mass of the point masses lying on the \(x\) -axis. $$ \begin{array}{l} m_{1}=12, m_{2}=1, m_{3}=6, m_{4}=3, m_{5}=11 \\ x_{1}=-6, x_{2}=-4, x_{3}=-2, x_{4}=0, x_{5}=8 \end{array} $$
Problem 28
Use the disk or the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about each given line. \(y=\frac{10}{x^{2}}, \quad y=0, \quad x=1, \quad x=5\) (a) the \(x\) -axis (b) the \(y\) -axis (c) the line \(y=10\)
Problem 28
(a) use a graphing utility to graph the region bounded by the graphs of the equations, \((b)\) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results. $$ y=x^{4}-2 x^{2}, \quad y=2 x^{2} $$
